User:Mkelly86/The Beurling-Selberg Extremal Problem

The Beurling–Selberg Extremal Problem is a problem in harmonic analysis that is principally motivated by its applications in number theory. Loosely speaking the problem asks: given a function $f:\mathbb {R} \rightarrow \mathbb {R}$, find a function $g$ with the properties that: $g$ is an entire function with controlled growth, $g(x)$ is real whenever $x$ is real, and the area between the graphs of $f(x)$ and $g(x)$ is as small as possible. Oftentimes an additional restriction on $g$ is: $f(x)\leq g(x)$ or $g(x)\leq f(x)$, i.e. $g(x)$ either majorizes or minorizes $f(x)$.

The subject was initiated with the unpublished work of Arne Beurling in the late 1930's and continued with Atle Selberg in the mid 1970's who used his results to prove a sharp form of the large sieve. Other notable applications include: improved bounds of the Riemann zeta function in the critical strip, Erdös–Turán inequalities, estimates of Hermitian forms, and a simplified proof of Montgomery and Vaughan's version of Hilbert's inequality .

The statement of the problem

The Beurling-Selberg extremal problem can be formulated in several different ways. Here we include several common formulations.

Majorant/minorant/best approximation

The Problem of Best Approximation Given $\delta >0$ and a function $f:\mathbb {R} \rightarrow \mathbb {R}$, find an analytic function $g:\mathbb {C} \rightarrow \mathbb {C}$ such that

1. $g(x)$ is real when $x\in \mathbb {R}$.
2. $g$ and $h$ are entire functions of exponential type at most $\delta$.
3. $f(x)-g(x)$ is minimized with respect to the $L^{1}(\mathbb {R} )$ norm.

Such a function $g$ is then called a best approximation to $f$. If in addition $f(x)\leq g(x)$ for every $x\in \mathbb {R}$, then the problem is the majorant problem and the function $g$ is an extremal majorant of $f$. Similarly, if $g(x)\leq f(x)$, then the problem is the minorant problem and the function $g$ is an extremal minorant of $f$. We call any of the solutions to the above problems Beurling–Selberg extremal functions of $f$. In applications it is often desirable to solve the majorant and minorant problem simultaneously, but simultaneous solutions need not exist. For instance, ${\text{log}}|x|$ has a known extremal majorant, but no extremal minorant of ${\text{log}}|x|$ exists because it would necessarily have a pole at zero.

Reformulation in a de Branges space

Let $E:\mathbb {C} \rightarrow \mathbb {C}$ be an entire function of bounded type in the upper half plane, and $|E(x-iy)|<|E(x+iy)|$ for every $x\in \mathbb {R}$ and $y>0$. Every such function $E(z)$ has an associated de Branges Space which we will denote $H(E)$ with norm $\|\cdot \|_{E}$. In this formulation one wishes to obtain a majorant and minorant of some prescribed exponential type $\delta >0$. Generally speaking, the function one wishes to majorize or minorize is not analytic, so in contrast to the above problem, one (roughly) seeks to minimize the difference of the majorant and minorant with respect to $\|\cdot \|_{E}$. Of course, such a minimization can only occur if the difference is analytic.
Here is the formulation of the problem:

Given a function $h:\mathbb {R} \rightarrow \mathbb {R}$, determine functions $f$ and $g$ such that

1. $f$ and $g$ are of entire functions of exponential type at most $\delta$
2. $f(x)\leq h(x)\leq g(x)$ for every $x\in \mathbb {R}$
3. $\displaystyle \int _{-\infty }^{\infty }\{g(x)-f(x)\}|E(x)|^{-2}dx$ is as small as possible.

A Simple Application

To demonstrate the utility of the Beurling-Selberg extremal functions, we consider the problem of estimating

$\displaystyle \int _{\alpha }^{\beta }|f(x)|^{2}dx$

where $\alpha <\beta$ and $f(x)$ is an almost periodic function

$f(x)=\displaystyle \sum _{n=1}^{N}a(n)e^{2\pi i\lambda _{n}x}$

where $\lambda _{1},...,\lambda _{N}$ are real numbers such that $\lambda _{i}\neq \lambda _{j}$ when $i\neq j$ and $a(1),...,a(N)$ are complex numbers.
Let $\delta =\min\{|\lambda _{i}-\lambda {j}|:i\neq j\}>0$ and let $C_{[\alpha ,\beta ]}(z;\delta )$ and $c_{[\alpha ,\beta ]}(z;\delta )$ be the Beurling-Selberg extremal majorant and minorant of $\chi _{[\alpha ,\beta ]}(x)$ of exponential type $2\pi \delta$. To simplify notation let $C(z)=C_{[\alpha ,\beta ]}(z;\delta )$ and $c(z)=c_{[\alpha ,\beta ]}(z;\delta )$, then

$c(x)\leq \chi _{[\alpha ,\beta ]}(x)\leq C(x)$

for every real $x$. Observe

$\displaystyle \int _{\alpha }^{\beta }|f(x)|^{2}dx\leq \displaystyle \int _{-\infty }^{\infty }C(x)|f(x)|^{2}dx$

but after writing $|f(x)|^{2}=f(x){\overline {f(x)}}$ and rearranging we get

$\displaystyle \int _{-\infty }^{\infty }C(x)|f(x)|^{2}dx=\displaystyle \sum _{n=1}^{N}\displaystyle \sum _{m=1}^{N}a(n){\overline {a(m)}}\displaystyle \int _{-\infty }^{\infty }C(x)e^{2\pi i(\lambda _{n}-\lambda _{m})x}dx$

But

$\displaystyle \int _{-\infty }^{\infty }C(x)e^{2\pi i(\lambda _{n}-\lambda _{m})x}dx={\hat {C}}(\lambda _{n}-\lambda _{m})$

By the Paley-Wiener theorem ${\hat {C}}(t)=0$ if $|t|\geq \delta$, thus

$\displaystyle \int _{\alpha }^{\beta }|f(x)|^{2}dx\leq {\hat {C}}(0)\displaystyle \sum _{n=1}^{N}|a(n)|^{2}$.

By repeating the same argument with $c(z)$ we obtain the estimate

${\hat {c}}(0)\displaystyle \sum _{n=1}^{N}|a(n)|^{2}\leq \displaystyle \int _{\alpha }^{\beta }|f(x)|^{2}dx\leq {\hat {C}}(0)\displaystyle \sum _{n=1}^{N}|a(n)|^{2}$.

Now using the fact that ${\hat {C}}(0)=\beta -\alpha +\delta ^{-1}$ and ${\hat {c}}(0)=\beta -\alpha -\delta ^{-1}$ the estimate can finally be rewritten as

$\displaystyle \int _{\alpha }^{\beta }|f(x)|^{2}dx=(\beta -\alpha +\theta \delta ^{-1})\displaystyle \sum _{n=1}^{N}|a(n)|^{2}$

where $-1<\theta <1$. This identity was also obtained by Montgomery and Vaughan from a generalization of Hilbert's inequality.

The Problem for the Signum function

The Interpolation Approach

$B(x)$ is shown here in red, ${\text{sgn}}(x)$ in black, and the extremal minorant $H(x)-K(x)$ in blue.

In the late 1930's Beurling considered the majorant problem for the signum function:

${\text{sgn}}(x)={\begin{cases}1&{\text{ if }}x>0\\0&{\text{ if }}x=0\\-1&{\text{ if }}x<0\\\end{cases}}$

for which he obtained the solution:

$B(z)=\left({\dfrac {\sin(\pi z)}{\pi }}\right)^{2}\left\lbrace \displaystyle \sum _{n=0}^{\infty }(z-n)^{-2}-\displaystyle \sum _{m=-\infty }^{-1}(z-m)^{-2}+2z^{-1}\right\rbrace .$

Furthermore he showed that $B(z)$ is unique in the sense that if $F(z)$ is another entire function of exponential type $2\pi$, and ${\text{sgn}}(x)\leq F(x)$, then

$\displaystyle \int _{-\infty }^{\infty }F(x)-{\text{sgn}}(x)\,dx\geq 1$

$H(x)$ is shown here in blue, ${\text{sgn}}(x)$ is in black.

with equality if and only if $F(z)=B(z)$.

Observe that the odd part of $B(z)$ is given by

$H(z)=\left({\dfrac {\sin(\pi z)}{\pi }}\right)^{2}\left\lbrace \displaystyle \sum _{n=-\infty }^{\infty }{\text{sgn}}(n)(z-n)^{-2}+2z^{-1}\right\rbrace$

and the even part of $B(z)$ is given by

$K(x)$ is shown here in blue.

$K(z)=\left({\dfrac {\sin(\pi z)}{\pi z}}\right)^{2},$

which is Fejér's Kernel for $\mathbb {R}$. It can be shown that

$H(x)-K(x)\leq {\text{sgn}}(x)\leq H(x)+K(x)=B(x)$

and $H(z)-K(z)$ is the extremal minorant.
The solution for the problem of best approximation is also known and is given by:

$G(z)=\left({\dfrac {\sin(\pi z)}{\pi }}\right)\left\lbrace \displaystyle \sum _{n\neq 0}(-1)^{n}{\text{sgn}}(n)\{(z-n)^{-1}+n^{-1}\}+\log(4)\right\rbrace .$

Minimization in a de Branges space

If $E(z)$ is an entire function of Bounded Type in the upper half plane, and $|E(x-iy)|<|E(x+iy)|$ for every $x\in \mathbb {R}$ and $y>0$, the Beurling-Selberg extremal functions (with the minimization taking place in $H(E)$) for ${\text{sgn}}(x)$ are known.
Let $\xi$ be a real number such that $E(\xi )\neq 0$. If $K(\omega ,z)$ is the reproducing kernel for $H(E)$ define $k_{\xi }(z)$ by

$k_{\xi }(z)={\dfrac {K(\xi ,z)}{K(\xi ,\xi )}}.$

Corresponding to $k_{\xi }(z)$ is an associated function $\ell _{\xi }(z)$ which is initially defined in a strip, but can be shown to extend to an entire function by analytic continuation, given by

$\ell _{\xi }(z+\xi )=k_{\xi }(z+\xi )^{2}\displaystyle \int _{-\infty }^{\infty }{\text{sgn}}(u)\left(1-e^{-2\pi zu}\right)d\mu _{\xi }*\mu _{\xi }(u)$

where $\mu _{\xi }$ is the unique Borel probability measure that satisfies

$k_{\xi }(z+\xi )^{-1}=\displaystyle \int _{-\infty }^{\infty }e^{-2\pi zu}\,d\mu _{\xi }(u)$

in an open vertical strip that contains 0. The functions $k_{\xi }(z)$ and $\ell _{\xi }(z)$ can be shown to satisfy

$\ell _{\xi }(x)-k_{\xi }(x)^{2}\leq {\text{sgn}}(x-\xi )\leq \ell _{\xi }(x)+k_{\xi }(x)^{2}$

and if $T_{\xi }(z)$ and $S_{\xi }(z)$ are functions of exponential type less than or equal to twice the exponential type of $E(z)$ that satisfy $S_{\xi }(x)\leq {\text{sgn}}(x-\xi )\leq T_{\xi }(x)$, then

${\dfrac {1}{K(\xi ,\xi )}}\leq {\dfrac {1}{2}}\displaystyle \int _{-\infty }^{\infty }\{T_{\xi }(x)-S_{\xi }(x)\}|E(x)|^{-2}dx$

with equality if and only if

$T_{\xi }(z)=\ell _{\xi }(x)-k_{\xi }(x)^{2}$

and

$S_{\xi }(z)=\ell _{\xi }(x)+k_{\xi }(x)^{2}.$

The problem in several variables

Compared to what is known in the single variable case, relatively little is known about the Beurling-Selberg extremal problem for several variables. Selberg developed a procedure to majorize and minorize a box in Euclidean space whose sides are parallel to to the coordinate axis. It is easy to construct a majorant of such a function by multiplying the known majorants of characteristic functions of intervals. A minorant is less simple and can be obtained as the combination of majorants and minorants, the periodic case is treated in the paper of Barton, Montgomery, Vaaler.

Characteristic function of a ball in Euclidean space

The only known function for which the Beurling-Selberg extremal problem has been solved in several variables is the characteristic function of the ball of radius $r$ and center 0 in $\mathbb {R} ^{n}$, which we will denote$\chi _{r}$:

$\chi _{r}(x)={\begin{cases}1&{\text{if }}\|x\|<r\\1/2&{\text{if }}\|x\|=r\\0&{\text{if }}\|x\|>r.\end{cases}}$

In particular, for fixed $\nu >-1$, $r>0$, and $\delta >0$, they find an explicit majorant $G$ and minorant $F$ that have exponential type at most $2\pi \delta$ and minimize the value of the integral

$\displaystyle \int _{\mathbb {R} ^{n}}\{G(x)-F(x)\}|x|^{2\nu +2-n}\,dx.$

We will let $H_{\nu }^{(n)}(r,\delta )$ denote the minimum value of this integral. In order to solve the problem in $\mathbb {R} ^{n}$ they first solve the problem in $\mathbb {R}$ and radially extend the 1-dimensional solutions (which they show are extremal). Let $\chi _{[\alpha ,\beta ]}$ be the normalized characteristic function of the interval $[\alpha ,\beta ]$:

$\chi _{[\alpha ,\beta ]}(x)={\begin{cases}1&{\text{ if }}\alpha <x<\beta \\1/2&{\text{ if }}x=\alpha ,\beta \\0&{\text{ if }}x>\beta {\text{ or }}x<\alpha .\\\end{cases}}$

then

$\chi _{[\alpha ,\beta ]}(x)={\dfrac {1}{2}}\left\lbrace {\text{sgn}}(x-\alpha )-{\text{sgn}}(x-\beta )\right\rbrace$.

Using the solution to the above problem for signum, the authors obtain the majorant and minorant as a linear combination of the majorant and minorant of the problem for the signum function:

${\dfrac {1}{2}}\left\lbrace S_{\alpha }(x)-T_{\beta }(x)\right\rbrace \leq \chi _{[\alpha ,\beta ]}(x)\leq {\dfrac {1}{2}}\left\lbrace T_{\alpha }(x)-S_{\beta }(x)\right\rbrace$.

The minimization occurs in a de Branges space that is in sympathy with radial extensions: a (de Branges) homogeneous space $H(E_{\nu })$ where $E_{\nu }(z)=A_{\nu }(z)-iB_{\nu }(z)$ and

$A_{\nu }(z)=\Gamma (\nu +1)\left({\dfrac {1}{2}}z\right)^{-\nu }J_{\nu }(z)$

and

$B_{\nu }(z)=\Gamma (\nu +1)\left({\dfrac {1}{2}}z\right)^{-\nu }J_{\nu +1}(z).$

The following identity makes this choice of de Branges space clear:

$\|f\|_{E_{\nu }}^{2}=\displaystyle \int _{-\infty }^{\infty }|f(x)E_{\nu }(x)^{-1}|^{2}dx=c_{\nu }\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}|x|^{2\nu +1}dx.$

where $c_{\nu }=\pi 2^{-2\nu -1}\Gamma (\nu +1)^{-2}$ and $J_{\nu }$ is a Bessel function of the first kind.

For every $r>0,\nu >-1/2$ and $\delta =\pi ^{-1}$, $H_{\nu }^{(n)}(r,\pi ^{-1})$ satisfies the following inequality

$H_{\nu }^{(n)}(r,\pi ^{-1})\leq 2\omega _{n-1}r^{2\nu +1}\{rJ_{\nu }(r)^{2}+rJ_{\nu +1}(r)^{2}-(2\nu +1)J_{\nu }(r)J_{\nu +1}(r)\}^{-1}$

where $\omega _{n-1}$ is the surface area of n-sphere and equality occurs if and only if

$J_{\nu }(\pi \delta r)J_{\nu +1}(\pi \delta r)=0$.

The Problem in the Periodic Case

The Beurling-Selberg extremal problem has a natural analogue for periodic functions. The best approximation problem is:
Given a function $g:\mathbb {R} \rightarrow \mathbb {R}$ that is periodic with period 1, find an entire function $f(z)$ such that

• $f(x)$ is real whenever $x\in \mathbb {R}$
• $f$ is a trigonometric polynomial of degree at most $N$
• $f-g$ is as small as possible in the $L^{1}([0,1])$-norm.
  

If in addition $g(x)\leq f(x)$ for all $x\in \mathbb {R}$, the problem is the majorant problem. If $f(x)\leq g(x)$ for all $x\in \mathbb {R}$, the problem is the minorant problem.
Periodic analogues of problems on $\mathbb {R}$ can intuitively be approached by periodization of the non-periodic problem and then an application of the Poisson summation formula. While this idea is oftentimes in the background, there are some technicalities. For instance, Montgomery (1994) provides a method of solving the problem for the sawtooth function:

$\psi (x)={\begin{cases}\{x\}-1/2&{\text{ if }}x\neq 0\\0&{\text{ otherwise }}\end{cases}}$

($\{x\}$ is the fractional part of $x$)that avoids using the Poisson summation formula as was used in Vaaler (1985). The technicality in this case is the analogue for $\psi$ in $\mathbb {R}$ is ${\text{sgn}}(x)$ which is not absolutely integrable, so the Fourier transform is not immediately defined. Vaaler worked around the issue by writing $B(x)=H(x)+K(x)$ (defined above) and computing the Fourier transforms of ${\text{sgn}}(x)-H(x)$ and $K(x)$.

Functions for which the Beurling-Selberg Functions are known

There are several papers where it is shown how to produce the Beurling-Selberg extremal functions for a large class of functions. For instance, Vaaler & Graham (1981) took steps for majorizing and minorizing integrable functions with some additional regularity. In Vaaler (1985) it is shown how to majorize and minorize a function of bounded variation, and in Carneiro & Vaaler (2010) it is shown how to solve the problem of best approximation of functions of the form

$f_{\mu }(x)=\displaystyle \int _{0}^{\infty }\{\exp(-\lambda |x|)-\exp(\lambda )\}d\mu (\lambda )$

where $\mu$ is a Borel measure that satisifies

$\displaystyle \int _{0}^{\infty }{\dfrac {\lambda }{\lambda ^{2}+1}}d\mu (\lambda )<\infty .$

Examples of such functions include: $exp\{-\lambda |x|\}$, $\log |x|$ and $|x|^{\alpha }$ where $|\alpha |<1$.

The following table contains functions for which the Beurling-Selberg extremal problem has been worked out, and is far from complete. The references in the following table may not be the reference in which the functions were introduced, but rather serve as a source to find the functions explicitly.

Function	References
${\text{sgn}}(x)$	Beurling (unpublished) Vaaler (1985) Holt and Vaaler (1996)
$\chi _{[\alpha ,\beta ]}(x)$ When $\beta -\alpha \in \mathbb {Z}$	Selberg (lectures in mid-70's)(collected works - 1991) Holt and Vaaler (1996)
$\chi _{[\alpha ,\beta ]}(x)$ When $\beta -\alpha \notin \mathbb {Z}$	Logan(1977)
$\chi _{B_{r}(0)}(x_{1},...x_{n})$ Where $B_{r}(0)=\{\mathbf {x} \in \mathbb {R} ^{n}:\|x\|<r\}$	Holt and Vaaler (1996)
$\log |x|$	Lerma (1998 - Phd. Dissertation) Carniero, Vaaler (2010)
$e^{-\lambda |x|}$ for $\lambda >0$	Carniero, Vaaler (2010)
${\dfrac {2\lambda }{\lambda ^{2}+4\pi ^{2}x^{2}}}$ for $\lambda >0$	Carniero, Littmann, Vaaler (2010)
$e^{-\lambda \pi x^{2}}$ for $\lambda >0$	Carniero, Littmann, Vaaler (2010)
$e^{-\lambda \pi |x|^{r}}$ for $\lambda >0$ and $0<r\leq 1$	Carniero, Littmann, Vaaler (2010)
$(x^{2}+\lambda ^{2})^{-\beta }$ for $\lambda >0$ and $\beta >0$	Carniero, Littmann, Vaaler (2010)
$-\log \left({\dfrac {x^{2}+\alpha ^{2}}{x^{2}+\beta ^{2}}}\right)$ for $0<\alpha <\beta$	Carniero, Littmann, Vaaler (2010)
$|x|^{\sigma }$ for $\sigma >-1$ and $\sigma \neq 0,2,4,\dots$	Carniero, Vaaler (2010) Carniero, Littmann, Vaaler (2010) [note $\sigma >0$ for the minorant problem]
$-\log(x^{2}+\alpha ^{2})$ for $\alpha \geq 0$	Carniero, Littmann, Vaaler (2010)
$(-1)^{n+1}x^{2n}\log(x^{2})$ for $n\in \mathbb {N}$	Carniero, Littmann, Vaaler (2010)

Graphs of Some Known extremal functions

The following extremal functions have exponential type $2\pi$

• 
  The extremal functions for sgn(x)
• 
  The extremal functions for $\chi _{[-1,1]}$
• 
  The Gaussian $e^{-\pi x^{2}}$ is shown in gray, an extremal majorant in red and an extremal minorant in blue.
• 
  $\log |x|$ is shown in black and an extremal majorant in red.
• 
  H(x) (defined above)
• 
  K(x) (defined above)

See Also

• Atle Selberg
• Arne Beurling
• Bounded Type
• exponential type
• de Branges space
• Hilbert's inequality
• Erdös–Turán inequalities
• Riemann zeta function

References

• Barton, Jeffrey; Montgomery, Hugh; Vaaler, Jeffrey (2001). "Note on a Diophantine inequality in several variables". Proc. Amer. Math. Soc. 129 (2): 337–345 (electronic). doi:10.1090/S0002-9939-00-05795-6. ISSN 0002-9939. S2CID 118982870.
• Boas, Jr., Ralph Philip (1954). Entire functions. Academic Press Inc.. pp. 124-132.
• Carneiro, Emmanuel; Chandee, Vorrapan (2011). "Bounding $\zeta$ in the Critical Strip". J. Number Theory (N.S.). 131 (3): 363–384. doi:10.1016/j.jnt.2010.08.002. S2CID 119591029.
• Carniero, E.; Littmann, F.; Vaaler, J. (2010). "Gaussian subordination for the Beurling-Selberg extremal problem". Transactions of the American Mathematical Society. v1. 365 (7): 3493–3534. arXiv:1008.4969. doi:10.1090/S0002-9947-2013-05716-9. S2CID 50680870.

• Carneiro, Emanuel; Vaaler, Jeffrey D. (2010). "Some extremal functions in Fourier analysis. III". Constr. Approx. 31 (2): 259–288. doi:10.1007/s00365-009-9050-6. ISSN 0176-4276. S2CID 16770439.
• de Branges, Louis (1968). Hilbert spaces of entire functions. Prentice–Hall Inc. pp. ix+326. {{cite book}}: Unknown parameter |address= ignored (|location= suggested) (help)
• Drmota, Michael and Tichy, Robert F. (1997). Sequences, discrepancies and applications. Lecture Notes in Mathematics. Vol. 1651. Springer-Verlag. pp. xiv+503. ISBN 3-540-62606-9. {{cite book}}: Unknown parameter |address= ignored (|location= suggested) (help)CS1 maint: multiple names: authors list (link)
• Graham, S. W. and Vaaler, Jeffrey D. (1981). "A class of extremal functions for the Fourier transform". Transactions of the American Mathematical Society. 265 (1): 283–302. doi:10.2307/1998495. ISSN 0002-9947. JSTOR 1998495.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Holt, Jeffrey J. and Vaaler, Jeffrey D. (1996). "The Beurling-Selberg extremal functions for a ball in Euclidean space". Duke Math. J. 83 (1): 202–248. doi:10.1215/S0012-7094-96-08309-X. ISSN 0012-7094.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Logan, B.F. (1977). "Bandlimited functions bounded below over an interval". Notices Amer. Math. Soc. 24: A-331.
• Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conference Series in Mathematics. Vol. 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC. pp. xiv+220. ISBN 0-8218-0737-4.
• Montgomery, Hugh L. (1978). "The analytic principle of the large sieve". Bull. Amer. Math. Soc. 84 (4): 547–567. doi:10.1090/S0002-9904-1978-14497-8. ISSN 0002-9904. {{cite journal}}: Unknown parameter |fjournal= ignored (help)
• Montgomery, H. L.; Vaughan, R. C. (1974). "Hilbert's inequality". J. London Math. Soc. (2). 8: 73–82. doi:10.1112/jlms/s2-8.1.73. ISSN 0024-6107.
• Selberg, Atle (1991). Collected papers. Vol. II. Springer-Verlag. pp. viii+253. ISBN 3-540-50626-8. {{cite book}}: Unknown parameter |address= ignored (|location= suggested) (help); Unknown parameter |note= ignored (help)

• Vaaler, Jeffrey D. (1985). "Some extremal functions in Fourier analysis". Bull. Amer. Math. Soc. (N.S.). 12 (2): 183–216. doi:10.1090/S0273-0979-1985-15349-2. ISSN 0273-0979.

[[Category:Complex analysis]] [[Category:Harmonic analysis]] [[Category:Mathematical analysis]] [[Category:Fourier analysis]] [[Category:Number theory]] [[Category:Analytic number theory]] [[Category:Approximation theory]] [[Category:Inequalities]]